Birthday Paradox Calculator
Enter the number of people to find the probability that at least two share a birthday. 23 people already exceeds 50%, 50 people reach 97%!
Input Data
Results
At a glance:The Birthday Paradox studies the probability that at least two people in a group share a birthday. Intuitively, exceeding 50% seems to need about 183 people (half the days in a year), but only 23 suffice. The reason is all pairwise combinations: 23 people form C(23,2)=253 pairs, and any matching pair counts. The probability all n birthdays are distinct is P(distinct)=365!÷[(365−n)!×365ⁿ], so the match probability is 1−P(distinct).
Formula
Let n = people, d = days per year:
P(all distinct) = d! ÷ [(d−n)! × dⁿ]
P(at least one match) = 1 − P(all distinct)
Example: n=23, d=365 → P ≈ 50.7%
$$P(\text{match}) = 1 - \frac{d!}{(d-n)!\,d^n}$$How to Use
- Enter the group size n.
- Choose days per year (365 or 366).
- The tool computes the distinct and match probabilities.
- Probability grows faster with more people; at 50 it reaches 97%.
Case Studies
Classroom and office
A class of 30 students: match probability ≈ 70.6%.
A 50-person office: probability ≈ 97%, almost certainly someone shares a birthday.
FAQ
Why only 23 people for >50%?
Because the number of pairwise comparisons grows quadratically; 23 people already make 253 pairs.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.